Geoscience Reference
In-Depth Information
argument based on the conservation of PV. In this exper-
iment the flow is ageostrophic enough for the incoming
fluid parcels to climb over the plateau. In order to con-
serve PV, the decrease in depth should be compensated
by decreasing either the Coriolis parameter or the relative
vorticity. The Coriolis parameter is lower southward such
that the flow can turn toward the equator (topographic
steering). However, in our experiment topographic steer-
ing is not obvious and the main effect is the generation of
anticyclonic relative vorticity. The parcel with induced cir-
culation experiences the Coriolis force, which is greater at
the northern side of the parcel due to the larger value of
theCoriolis parameter. The resulting totalforceis a restor-
ing force that drives the (anticyclonic) parcel northward.
The parcel moving northward then acquires cyclonic vor-
ticity and is moved southward and the process repeats
itself. This is the basic mechanism of the generation of
a Rossby wave.
It is useful to check if the observed wave pattern is con-
sistent with the dispersion relation (5.26) which can be
simply modified to include an incoming stream of angular
velocity
:
spatially localized sources/forcing on the
β
-plane [
Stom-
mel
, 1982;
Davey and Killworth
, 1989;
Rhines
, 1994].
Experimental demonstration of
β
-plumes on the polar
β
plane was provided in the work of
Afanasyevetal.
[2012].
An important dimensionless parameter in the problem
of flows over topography is
R
β
=
βd
2
/U
[
Lighthill
, 1967;
Rhines
, 2007], which is the square of the ratio of the
width (north-south) of the mountain to the zonal length
scale of the stationary Rossby wave speed. To calculate
R
β
, we have to consider the flow in the
β
-plane context
rather than in the context of the polar
β
-plane as we have
been doing so far. In our experiment,
R
β
= 0.5 is rela-
tively small, which indicates that the mountain is not wide
enough and the current is too strong for blocking to occur.
A classic lee Rossby wave is observed instead.
The flow is strikingly different if the mean flow is west-
ward (rather than eastward as in the previous example).
The westward mean flow is induced by accelerating the
tank slightly above the null rate to
= 1.02
0
. The per-
turbation in the lee of the mountain is now in the form
of a train of inertial waves (indicated by 1 in Figure 5.4).
The Inset in Figure 5.4a shows the relative vorticity of the
flow in the lee of the mountain while Figure 5.4b shows
isolines of the surface elevation
η
. The displacement of
the isolines of
η
indicates that the amplitude of the inertial
waves is about 2
μ
m, which is quite small. The wavelength
can be easily estimated by measuring the distance between
the crests of the wave. It is noticeably shorter than that of
the Rossby wave in the eastward flow. Just as we did with
Rossby waves, we can perform a consistency check of the
observations with the appropriate dispersion relation. For
stationary waves we assume again that their frequency is
ω
=
kU
. For measured values of the wavelength of the
wave of about 4 cm and the velocity of the mean flow at
the latitude of the mountain of 0.5 cm/s, we obtain the
dimensionless frequency
ω/f
0
= 0.17. This value is shown
by the circle in Figure 5.2a and is in good agreement with
the dispersion relation (5.20). Note that we can also cal-
culate the vertical wave number
γ
n
using (5.21). The value
of the product
γ
n
H
0
=2.7
<π
indicates that this is mode
n
= 0, which has the most simple vertical structure.
The flow shown in Figure 5.4 is actually more com-
plicated than a simple lee wave. Wave breaking can
be observed in the region indicated by 2 in Figure 5.4.
This region is a shear layer formed by a relatively high
velocity current at the northern flank of the mountain.
This current is indicated by the concentrated isolines in
Figure 5.4b and most likely occurs due to the effect of
topographic steering. Indeed, the flow climbing the moun-
tain is deflected northward where the Coriolis parameter
is larger (or equivalently the depth is smaller) to con-
serve PV. Inertial waves break in the shear layer and create
disturbances that propagate downstream. Careful obser-
vation shows that these disturbances, in turn, emit inertial
2
mγ R
b
−
m
=
ω
mn
−
α
mn
R
b
/R
2
+1
.
(5.27)
The angular velocity of the mean flow is such that
=
0
−
initially. For stationary waves
ω
mn
=0.Asmall
mountain, not unlike a point source, excites an entire spec-
trum of waves with corresponding wave numbers
m
and
n
. A dominant mode, however, can be easily identified in
Figure 5.3 with a zonal wavelength (in angular measure)
of about 30
◦
, which corresponds to
m
= 12. The first zero
of the Bessel function is at
α
12,1
= 16.7, which gives the
zonal phase speed
ph
=
ω/m
=
0.024s
−
1
.Herethe
minus sign indicates westward propagation. This value is
close (and of opposite sign) to the mean rotation rate of
the fluid in the tank,
= 0.021s
−
1
measured at the same
time as when the wave pattern in Figure 5.3 was observed.
The reader should also refer to the high contrast images
of flows over bottom topography made with the original
gray scale altimetry by
Rhines et al.
(2007). They show
in particular an effect of upstream blocking caused by
very long Rossby waves that are fast enough to propagate
upstream from the mountain. These waves are described
by a simple solution of (5.27) for stationary waves given
by
m
= 0. Their speed is somewhat limited by dispersion,
which is represented by the term with
α
mn
such that the
fastest phase speed is achieved for the minimum value
α
01
= 2.4. These waves have zonal crests and form a pat-
tern of almost steady zonal currents such that there is
a stagnant region upstream of the mountain. These cur-
rents are in fact
β
-plumes, not unlike those generated by
−
